Step-by-step explanation:
Using Scientific Notation
Recall at the beginning of the section that we found the number \displaystyle 1.3\times {10}^{13}1.3×10
13
when describing bits of information in digital images. Other extreme numbers include the width of a human hair, which is about 0.00005 m, and the radius of an electron, which is about 0.00000000000047 m. How can we effectively work read, compare, and calculate with numbers such as these?
A shorthand method of writing very small and very large numbers is called scientific notation, in which we express numbers in terms of exponents of 10. To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between 1 and 10. Count the number of places n that you moved the decimal point. Multiply the decimal number by 10 raised to a power of n. If you moved the decimal left as in a very large number, \displaystyle nn is positive. If you moved the decimal right as in a small large number, \displaystyle nn is negative.
For example, consider the number 2,780,418. Move the decimal left until it is to the right of the first nonzero digit, which is 2.
The number 2,780,418 is written with an arrow extending to another number: 2.780418. An arrow tracking the movement of the decimal point runs underneath the number. Above the number a label on the number reads: 6 places left.
We obtain 2.780418 by moving the decimal point 6 places to the left. Therefore, the exponent of 10 is 6, and it is positive because we moved the decimal point to the left. This is what we should expect for a large number.
\displaystyle 2.780418\times {10}^{6}2.780418×10
6
Working with small numbers is similar. Take, for example, the radius of an electron, 0.00000000000047 m. Perform the same series of steps as above, except move the decimal point to the right.
The number 0.00000000000047 is written with an arrow extending to another number: 00000000000004.7. An arrow tracking the movement of the decimal point runs underneath the number. Above the number a label on the number reads: 13 places right.
Be careful not to include the leading 0 in your count. We move the decimal point 13 places to the right, so the exponent of 10 is 13. The exponent is negative because we moved the decimal point to the right. This is what we should expect for a small number.
\displaystyle 4.7\times {10}^{-13}4.7×10
−13
A GENERAL NOTE: SCIENTIFIC NOTATION
A number is written in scientific notation if it is written in the form \displaystyle a\times {10}^{n}a×10
n
, where \displaystyle 1\le |a|<101≤∣a∣<10 and \displaystyle nn is an integer.
